one of the most important concepts of mathematics, answering the need to find functions given their derivatives (for example, to find the function expressing the path traversed by a moving point given the velocity of that point), on the one hand, and to measure areas, volumes, lengths of arcs, the work done by forces in a given interval of time, and so forth, on the other. These problems involve so-called indefinite integrals and definite integrals. Their computation is the task of the integral calculus.
Indefinite integral. An antiderivative of a function f (x) of a single real variable is a function F (x) whose derivative for each value of (x) is equal to f(x). By adding a constant to an antiderivative of a function, we again obtain an antiderivative of that function. Consequently, having one antiderivative F (x) of a function f (x), we obtain a general expression for all antiderivatives of that function in the form F (x) + C. This general expression for antiderivatives is called the indefinite integral
∫f(x) dx
of the function f(x). One of the fundamental theorems of the integral calculus establishes the fact that every continuous function f(x) of a real variable has an indefinite integral.
Definite integral. The definite integral of a function f(x) with lower limit a and upper limit b may be defined as the difference
where F(x) is an antiderivative of the function f(x). This definition does not depend on which antiderivative is chosen for the computation of the definite integral. If the function f(x) is continuous, the above definition is equivalent for the case a < b to the following definition given by A. Cauchy (1823): Consider an arbitrary partition of the interval [a, b] determined by the points
(2) a = x0 < X1 < … < xn = b
In each subinterval [x11,Xi] (i = 1, 2, …, n) we take an arbitrary point ξ i (i 1 ≤ ξ i ≤ xi) and form the sum
(3) Sn = f(ξ1)(x1-x0)+f(ξ2)(x2-x1)+ … + f(ξn)(xn-xn_1)
The sum Sn depends on the choice of the points xi, and ξi. However, in the case of a continuous function f(x), the sums Sn obtained by choosing different points xi and ξi, converge to the same limit provided that the maximum of the differences xi— Xi, 1 tends to zero as n—∞. This limit is precisely the definite integral
We define:
As noted above, the definite integral can be expressed by means of any antiderivative F(x). Conversely, an antiderivative F(x) can be written in the form
where a is an arbitrary constant. Accordingly, the indefinite integral can be written in the form
For the evolution of the concept of the integral and the properties of indefinite and definite integrals seeINTEGRAL CALCULUS.
Generalization of the concept of the integral.
Riemann integral Cauchy applied his definition of the integral only to continuous functions. B. Riemann (1853) defined the integral.
as the limit of the sums Sn as max(xi – xi_1)→ 0 in all those cases when this limit is uniquely defined. He also investigated the conditions of applicability of such a definition. H. Lebesgue gave (1902) a more complete form to these conditions using his concept of the measure of a set. Two conditions are necessary and sufficient for a function f(x) to be integrable over [a, b] in the sense of Riemann: f(x) must be bounded on [a,b], and the set of discontinuities of the function f(x) on [a,b] must have measure zero. Thus, continuity at each point of the interval [a,b] is not at all required for Riemann integrability.
It is now possible to define the antiderivative and the indefinite integral by means of formulas (4) and (5). However, it should be noted that in this case the antiderivative F(x) does not necessarily have the integrand f(x) as its derivative at every point. But at every point where f(x) is continuous, that is, by virtue of Lebesgue’s result, everywhere except perhaps on a set of measure zero, we have
G. Darboux (1879) gave a definition of the Riemann integral rendering the conditions for the existence of such an integral intuitively clear. In place of the sums (3) Darboux introduced the sums (called Darboux sums)
S̅ = M1(x1–x0) + M2(x2 – x1)+…+Mn(xn–xn-1)
S = m1(xl – x0) + m2(x2 - x1) + … + mn(xn–xn-1)
where Mk is the least upper bound of the function f(x) on the subinterval [xk _ 1,xk] and mk is the greatest lower bound of f(x) on that subinterval. If I is the greatest lower bound of the sums S̅ and I is the least upper bound of the sums S̲, then the condition ̲ I = Ī is necessary and sufficient for the existence of the Riemann integral. The common value I = Ī = I̅ of the numbers I̲ and Ī is the Riemann integral (6). The numbers Ī and I are called, respectively, the upper and lower Darboux integrals.
Lebesgue integral. The concept of the measure of sets introduced by Lebesgue permitted a significantly broader definition of the integral. In order to define the integral (6), Lebesgue divides the range of possible values of the variable y = f(x) by means of points
… < y-2 < y-i < y0 < … < yi < …
and designates as Mi the set of those points x in the interval [a, b] for which
yi-1 ≤ f(x) < yi
The sum S is defined by the equality
where ηi is selected from the interval yi–1 ≤ ηi, < yi, and μ (Mi) denotes the measure of Mi. The function f(x) is said to be integrable in the sense of Lebesgue on the interval [a,b] if the series that define the sums S converge absolutely as max(yi, – yi – 1) → 0. The limit of these sums is called the Lebesgue integral (6). One may define the antiderivative in the sense of Lebesgue as a function F(x) satisfying the equality (4), where the integral on the right is understood to be the Lebesgue integral. As in the case of the Riemann integral, equality (7) will in this case be fulfilled at all points except, perhaps, on a set of measure zero.
For a bounded function f(x) to be Lebesgue integrable it is necessary and sufficient that it belong to the class of functions measurable in the sense of Lebesgue. All functions that are encountered in mathematical analysis are measurable in this sense. Moreover, up to the present (1972) not a single specific example of a nonmeasurable function has been constructed. Thus, for the case of bounded functions, Lebesgue has solved the problem of defining the integral (6) with a generality that exhausts the requirements of mathematical analysis. Among Lebesgue integrable functions, there are infinitely many functions that are everywhere discontinuous and, consequently, not Riemann integrable. However, every Riemann integrable function is also Lebesgue integrable.
Lebesgue’s definition can be generalized to include integration over a half-line or the whole line, that is, to the case of integrals of the form
After this generalization, Lebesgue’s theory encompasses all cases of absolutely convergent improper integrals.
The generality attained in Lebesgue’s definition is essential for many problems of mathematical analysis; for example, only after the introduction of the Lebesgue integral was it possible to establish the Riesz-Fischer theorem in the theory of trigonometric series, which asserts that any series
for which
represents the function f(x), for which the coefficients an and bn have the values
(the integrals are Lebesgue integrals).
Stieltjes integral. At the end of the 19th century, the Riemann integral was generalized in a manner that differed completely from that resulting from the introduction of the concept of Lebesgue measure. This generalization was due to T. Stieltjes (1894). Specifically, let f(x) be a continuous function of a real variable x defined on the interval [a, b], and let U(x) be a bounded monotonic (nondecreasing or nonincreasing) function defined on the same interval. For the definition of the Stieltjes integral we take an arbitrary partition (2) of the interval [a,b] and form the sum
f(ξ1) [U(x1) – U(x0) + f(ξ2)[U(x2)– U(x1)] + … + f(ξn)[U(xn) – (U(xn-1)]
where ξ1, ξ2, …, ξn are arbitrary points chosen from the subintervals [x0, x1], [xi, x2], …, [xn – 1, xn], respectively. Let δ be the largest distance between two neighboring points of the partition (2). If we take any sequence of partitions for which δ tends to zero, then the sum (8) tends to be the same limit no matter which points ξ1, ξ2, …, are selected in the respective intervals. This limit is termed, following Stieltjes, the integral of the function f(x) with respect to the function U(x) and is denoted by the symbol
The integral (9) (also called the Stieltjes integral) exists even in the case when the bounded function U(x), while not itself mono-tonic, can be represented in the form of a sum or difference of two bounded monotonic functions U1(x) and U2(x):
U(x) = U1(x)– U2(x)
that is, when it is a function of bounded variation.
If the integrator U(x) has a bounded and Riemann integrable derivative U’(x), then the Stieltjes integral is related to the Riemann integral, namely,
In particular, when U(x) = x + C, the Stieltjes integral (9) becomes the ordinary Riemann integral (6).
Further generalizations. The concepts of the integral owing to Stieltjes and Lebesgue were subsequently successfully combined and generalized to yield a concept of integration over any (measurable) set in a space of any number of dimensions. Classical multiple integrals are fully covered by this approach. The requirements of disciplines such as probability theory and the general theory of dynamic systems have led to the still broader concept of the abstract Lebesgue integral, which is based on the general concepts of measure of sets and measurability of functions. Specifically, let X be a space with a system B of its subsets called measurable. It is assumed that B is closed under the usual set-theoretical operations performed finitely many or countably many times. Let μ, be a finite measure on B. For a 5-measurable function y = f(x), x ∊ X, assuming a finite or countable number of values y1, y2, …, yn, …, respectively, on the pairwise disjoint sets A1, …, An, …, whose sum is X, we define the integral of the function f(x) with respect to the measure μ, to be the sum of the series
with the assumption that this series converges absolutely. The integral just defined is denoted by the symbol
∫ xf(x)μ(dx)
For other f integrability and the integral are defined by a certain natural passage to the limit involving the “piecewise constant” functions introduced above.
Let A be a measurable set and let ϕ A(x) = 1 for x belonging to A and ϕ A(x) = 0 for x not belonging to A. Then the integral of f(x) on the set A is defined by
∫Af(x)μ (dx) = ∫ xf(x)ϕA(x)μ(dx)
When μ and A are fixed, the integral, viewed as a function of f, may be regarded as a linear functional; when f is fixed, the integral, viewed as a function of the set A, is a countably additive function.
It should be noted that despite its apparent abstractness this general concept of an integral is very well suited for the definition of such a concept as mathematical expectancy (in probability theory) and even for the general formulation of the testing of statistical hypotheses. Integrals with respect to the so-called Wiener measure and its various analogs are used in statistical physics (here X is the space of functions continuous on some interval). The generalizations of the concept of the integral mentioned so far have been such that f and f prove to be either both integrable or both nonintegrable.
Generalizations of the initial concept of the integral in another direction apply to functions of a single variable, but they contribute much more to the study of the integration of unbounded functions. Already Cauchy, in the case of a function f(x) unbounded at a point x = c, defined the integral
when a < c < b, as the limit of the expression
as ξ1 → 0 and ξ2 → 0. Analogously, the integral with infinite limits
is defined as the limit of the integral
as a → – ∞ and b → + ∞. If in this case the integrability of f(x) is not required, then this definition of Cauchy is not covered by Lebesgue’s definition.
A still broader generalization of the concept of the integral in this direction was proposed by A. Denjoy (1912) and A. Ia. Khinchin (1915).
REFERENCES
Lebesgue, H. Integrirovanie i otyskanie primitivnykh funktsii. Moscow-Leningrad, 1934. (Translated from French.)
Saks, S. Teoriia integrala. Moscow, 1949. (Translated from English.)
Kamke, E. Integral Lebega-Stilt’esa. Moscow, 1959. (Translated from German.)
Whitney, H. Geometricheskaia teoriia integrirovaniia. Moscow, 1966. (Translated from English.)
Rudin, W. Osnovy matematicheskogo analiza. Moscow, 1962. (Translated from English.)
Dunford, N., and J. T. Schwarz. Lineinye operatory: Obschaia teoriia. Moscow, 1962. (Translated from English.)
Neveau, J. Matematicheskie osnovy teorii veroiatnostei. Moscow, 1969. (Translated from French.)
Federer, H. Geometric Measure Theory. Berlin-Heidelberg-New York, 1969.
Under the editorship of AcademicianA. N. KOLMOGOROV
integral - the result of a mathematical integration; F(x) is the integral of f(x) if dF/dx = f(x)